Problem 56 - Algebra 1

Question

Solve the quadratic equation. $$x^{2}-10 x=0$$

Step-by-Step Solution

Answer

The roots of the quadratic equation \(x^{2}-10 x=0\) are \(x = 0\) and \(x = 10\).

Step 1: Identify the Form and Constants

The given equation, \(x^{2}-10 x=0\), is of the form \(ax^2 + bx + c = 0\). Here, \(a=1\), \(b=-10\), and \(c=0\).

Step 2: Factor out the Common Factor

In the quadratic equation, a common factor of \(x\) can be factored out. Factoring out the \(x\) from each term gives \(x(x - 10) = 0\).

Step 3: Set Each Factor Equal to Zero

In order to find the roots of the equation, set each factor equal to zero and solve for \(x\). This gives the following two equations: \(x = 0\) and \(x - 10 = 0\). Solving the second equation gives \(x = 10\).

Key Concepts

Factoring Quadratic Equations

Factoring is a method used to solve quadratic equations by expressing them as the product of two or more simpler expressions. A quadratic equation usually has the form \( ax^2 + bx + c = 0 \). The process starts with identifying the greatest common factor (GCF) of the terms, if there's any, and then applying techniques such as grouping or using formulas like the difference of squares.

For example, in the exercise \(x^2 - 10x = 0\), factoring involves taking out the GCF, which is \(x\), giving us \( x(x - 10) = 0 \). This simplifies the equation and sets the stage for us to use the zero product property to find the equation's roots.

Finding Roots of Quadratic Equations

The roots of a quadratic equation are the values of \(x\) that make the equation true when they replace \(x\) in the equation. The roots, also known as the solutions or zeroes, are where the graph of the equation crosses or touches the \(x\)-axis.

We can find these roots using various methods, including factoring, completing the square, the quadratic formula, or graphically. In the provided step-by-step solution, after factoring the equation \(x^2 - 10x = 0\), we set each factor equal to zero: \(x = 0\) and \(x - 10 = 0\), which yields the roots \(x = 0\) and \(x = 10\).

Quadratic Formula

The quadratic formula provides a straightforward way to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are coefficients from the equation, and \(\pm\) denotes that there are usually two solutions.

For the equation used in our example that's already factored, using the quadratic formula isn't necessary. However, in cases where factoring is difficult or impossible, the quadratic formula is an invaluable tool for finding roots.

Zero Product Property

The zero product property states that if a product of factors equals zero, then at least one of the factors must be zero. It's a critical concept when solving quadratic equations through factoring.

In the context of the provided solution, after factoring \(x^2 - 10x = 0 \) into \(x(x - 10) = 0\), we apply the zero product property. Setting each factor equal to zero --- \(x = 0\) and \(x - 10 = 0\) --- allows us to find the equation's roots. Understanding and applying this property is essential in the process of solving quadratic equations.